已知数列\(\{x_{n}\}\)满足\({{x}_{1}}=\dfrac{{1}}{{2}}\),\({{x}_{n+1}}=\dfrac{{1}}{{1}+{{x}_{n}}}\),\(n∈N^{*}\).
\((1)\)猜想数列\(\{x_{2n}\}\)的单调性,并证明你的结论:
\((2)\)证明:\(|{{x}_{n+1}}-{{x}_{n}}|\leqslant \dfrac{1}{6}{{\left( \dfrac{2}{5} \right)}^{n-1}}\).
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